I want to find the equation of the intersection between a sphere and cylinder (in the first octant) but it's kind of weird.

Sphere: x^2 + y^2 +z^2 = 4
Cylinder: x^2 + y^2 - 2y = 0

If I just sub one of them into the other I get:
2y + z^2 = 4

but that doesn't make sense to me since that is an equation of a surface since x can vary. There should be some sort of restriction on x... but how do I get this?

The intersection will be a curve, not a surface, and that is best described parametrically. Write the equation of the cylinder as , and you see that this can be parametrised as . Then , and if you substitute that into the equation of the sphere you see that .

Therefore the part of the curve in the positive octant can be described by the parametric representation .

The intersection will be a curve, not a surface, and that is best described parametrically. Write the equation of the cylinder as , and you see that this can be parametrised as . Then , and if you substitute that into the equation of the sphere you see that .

Therefore the part of the curve in the positive octant can be described by the parametric representation .